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I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some distributions depending on $N$. For the case $N=2$, a correlation matrix is like $$\begin{bmatrix}1&x\\x&1\end{bmatrix}$$ with $x\in[-1,1]$ and its eigenvalues are $1\pm x$ and so its largest eigenvalue has an upper bound $2$. Also one can see that the distribution of the largest eigenvalue is actually a uniform distribution on $[1,2]$, when this correlation matrix is sampled uniformly ($x$ uniform in $[-1,1]$).

One would be interested to know the analogue in general dimension, or even the limit $N\rightarrow\infty$, i.e. distribution or bounds for the largest eigenvalue. I looked through literatures and didn't find any exact answers. Is there any development on this topic? Or is it still an open problem?

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It sounds like you are asking for the distribution of the eigenvalues of a random symmetric matrix with i.i.d. centered entries. This is a huge subject, and you should read the oeuvre of Mehta, or Terry Tao's blog posts, or Anderson/Guionnet/Zeitouni, or..

Mehta, Madan Lal, Random matrices., Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier (ISBN 0-12-088409-7/hbk). xviii, 688 p. (2004). ZBL1107.15019.

Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer, An introduction to random matrices, Cambridge Studies in Advanced Mathematics 118. Cambridge: Cambridge University Press (ISBN 978-0-521-19452-5/hbk). xiv, 492 p. (2010). ZBL1184.15023.

ADDENDUM For random correlation matrices, it is a little less clear how you produce them, and the results depend on the model. A detailed discussion is given in:

Holmes, R. B., On random correlation matrices, SIAM J. Matrix Anal. Appl. 12, No. 2, 239-272 (1991). ZBL0723.15020.

(see section 3.2 and the sequel).

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    $\begingroup$ Thanks. I agree that RMT is most likely to answer this. However the off diagonal entries are actually not i.i.d. as the requirements for a matrix to be a "correlation matrix" is 1) diagonal entries are 1; 2) symmetric; 3) semi-positive definite. I believe the complexity is hiding behind the third requiement. $\endgroup$ – erachang Nov 6 '18 at 4:16
  • $\begingroup$ @erachang See the addendum. $\endgroup$ – Igor Rivin Nov 6 '18 at 4:41

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